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Geometry and Trigonometry Difficulty: Hard

The three points shown define a circle. The circumference of this circle is $k π$ , where $k$ is a constant. What is the value of $k$ ?

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Explanation

The correct answer is $8$ . It's given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point $left parenthesis 6 comma 6 right parenthesis$ is halfway between the points $left parenthesis 6 comma 2 right parenthesis$ and $left parenthesis 6 comma 10 right parenthesis$ , and is a distance of $4$ units from each of those two points. The point $left parenthesis 6 comma 6 right parenthesis$ is also a distance of $4$ units from $left parenthesis 2 comma 6 right parenthesis$ . Because the point $left parenthesis 6 comma 6 right parenthesis$ is the same distance from all three points shown, it must be the center of the circle. Since that distance is $4$ , it follows that the radius of the circle is $4$ . The circumference of a circle with radius $r$ is equal to $2 pi r$ . It follows that the circumference of the given circle is $2 pi left parenthesis 4 right parenthesis$ , or $8 pi$ . It's given that the circumference of the circle is $k π$ . Therefore, the value of $k$ is $8$ .